Properties

Label 2-800-1.1-c1-0-15
Degree $2$
Conductor $800$
Sign $-1$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s − 6·13-s − 2·17-s − 10·29-s + 2·37-s + 10·41-s − 7·49-s − 14·53-s − 10·61-s + 6·73-s + 9·81-s + 10·89-s − 18·97-s − 2·101-s + 6·109-s + 14·113-s + 18·117-s + ⋯
L(s)  = 1  − 9-s − 1.66·13-s − 0.485·17-s − 1.85·29-s + 0.328·37-s + 1.56·41-s − 49-s − 1.92·53-s − 1.28·61-s + 0.702·73-s + 81-s + 1.05·89-s − 1.82·97-s − 0.199·101-s + 0.574·109-s + 1.31·113-s + 1.66·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.617316516935557446365175892726, −9.206579341791145473157585005601, −8.025782606763976976877695514876, −7.38383189330491254934984805094, −6.27997548582365804353754428543, −5.37327377173791634385808352079, −4.46989807490289628768060263585, −3.13241440937412885242101141742, −2.12458086750050120053975349819, 0, 2.12458086750050120053975349819, 3.13241440937412885242101141742, 4.46989807490289628768060263585, 5.37327377173791634385808352079, 6.27997548582365804353754428543, 7.38383189330491254934984805094, 8.025782606763976976877695514876, 9.206579341791145473157585005601, 9.617316516935557446365175892726

Graph of the $Z$-function along the critical line